Question

rachel invests $500 in an annuity that offers an interest rate of 5% compounded quarterly for 10 years. what is the value of rachel's investment after 10 years?

A. $821.81
B. $1326.65
C. $566.14
D. $814.45

Answer 1

To compute the future value of Rachel's investment in an annuity, we can use the formula for compound interest since the investment is compounded quarterly. The formula for the future value \( FV \) of an investment is given by:

\[
FV = P \left(1 + \frac{r}{n}\right)^{nt}
\]

where:
- \( P \) is the principal amount (the initial investment),
- \( r \) is the annual interest rate (as a decimal),
- \( n \) is the number of times interest is compounded per year,
- \( t \) is the number of years the money is invested for.

In this case:
- \( P = 500 \)
- \( r = 0.05 \)
- \( n = 4 \) (since the interest is compounded quarterly)
- \( t = 10 \)

Plugging in the values:

\[
FV = 500 \left(1 + \frac{0.05}{4}\right)^{4 \times 10}
\]

Calculating step-by-step:

1. Calculate \( \frac{0.05}{4} \):
\[
\frac{0.05}{4} = 0.0125
\]

2. Calculate \( 1 + 0.0125 \):
\[
1 + 0.0125 = 1.0125
\]

3. Calculate \( 4 \times 10 \):
\[
4 \times 10 = 40
\]

4. Calculate \( (1.0125)^{40} \):
\[
(1.0125)^{40} \approx 1.643619
\]

5. Finally, calculate \( FV \):
\[
FV = 500 \times 1.643619 \approx 821.81
\]

Thus, the value of Rachel's investment after 10 years is approximately **$821.81**, which corresponds to option **A**.